MinT - Best Known (172, 172+57, s)-Nets in Base 3

Best Known (172, 172+57, s)-Nets in Base 3

(172, 172+57, 328)-Net over F₃ — Constructive and digital

Digital (172, 229, 328)-net over F₃, using

3¹ times duplication [i] based on digital (171, 228, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 57, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(172, 172+57, 967)-Net over F₃ — Digital

Digital (172, 229, 967)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+57, 43334)-Net in Base 3 — Upper bound on s

There is no (172, 229, 43335)-net in base 3, because

1 times m-reduction [i] would yield (172, 228, 43335)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 079918 705799 368275 890072 327068 055592 114349 933108 686016 920654 986958 121906 438057 163856 055780 538187 009237 323209 > 3²²⁸ [i]